The base of a solid $S$ is the region bounded by the graphs of the functions $f$ and $g$ shown below. $y$ $x$ $(a,c)$ $(b,d)$ $ g$ $ f$ Cross-sections perpendicular to the $x$ -axis are semi-circles. Which one of these integrals represents the volume of solid $S$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{\pi}8\int_a^b[g(x)-f(x)]^2dx$ (Choice B) B $\dfrac{\pi}2\int_c^d[g(y)-f(y)]^2dy$ (Choice C) C $\dfrac{\pi}2\int_a^b[g(x)-f(x)]^2dx$ (Choice D) D $\dfrac{\pi}8\int_c^d[g(y)-f(y)]^2dy$ (Choice E) E None of the above
Solution: The base of the solid is shaded in blue. Let's add a thin orange rectangle to depict a representative cross-section sitting on the base. The length of the green segment is $k$. $y$ $x$ $k$ $(a,c)$ $(b,d)$ $ g$ $ f$ Since each cross-section is perpendicular to the $x$ -axis, the independent variable is $x$. Given an $x$ -value, we can calculate the length $k$ from the functions $f$ and $g$ via the equation $k=g(x)-f(x)$. We can see from the graph that $x$ goes from $a$ to $b$. If $A$ denotes the area of each cross-section as a function of $x$, the volume $V$ of solid $S$ is $ V=\int_a^b A(x) \,dx$. To determine the area $A$ as a function of $x$, first express $A$ in terms of $k$. Since the semi-circular cross-section rests on the rectangle pictured above, the diameter of the semi-circle is $k$. The radius of the semi-circle is $k/2$. $\dfrac k2$ $k$ The area $A$ of the semi-circle is $A=\dfrac12\cdot\pi \left(\dfrac k2\right)^2=\dfrac\pi8k^2$. What is $A$ as a function of $x$ ? Let's substitute $k=g(x)-f(x)$ into $A=\dfrac\pi8k^2$ to get $A(x)=\dfrac\pi8[g(x)-f(x)]^2$. Can you express the volume $V$ of solid $S$ as a definite integral? The volume formula gives us the definite integral $\begin{aligned} V&=\int_a^b A(x) \,dx \\\\ &=\int_a^b \dfrac\pi8[g(x)-f(x)]^2 \,dx \\\\ &=\dfrac\pi8\int_a^b [g(x)-f(x)]^2 \,dx \end{aligned}$